A multi-carrier communication system, such as a Discrete Multi-Tone (DMT) system in the various types of Digital Subscriber Line, for example, asymmetric digital subscriber line (ADSL) and very high-speed digital subscriber line (VDSL) systems, carries an information bit stream from a transmitter to a receiver. The information bit stream is typically converted into a sequence of data symbols having a number of tones. Each tone may be a group of one or more frequencies defined by a center frequency and a set bandwidth. The tones are also commonly referred to as sub-carriers or sub-channels. Each tone acts as a separate communication channel to carry information between a local transmitter-receiver (transceiver) device and a remote transceiver device.
DMT communication systems, such as ADSL, may experience channel distortion that causes the data symbols to spread. Therefore, there are certain limitations to how close symbols can be lined up next to each other without the use of a channel distortion compensation technique. The effect of channel distortion can be minimized using a channel equalizer.
FIG. 1 illustrates a conventional DMT receiver. A channel equalizer is used to control the spread of the data symbols after going through the channel. A cyclic prefix (CP) may be employed in such systems to simplify channel equalization to minimize a source of cross channel interference. Generally, if the length of the channel impulse response is equal to or less than the cyclic prefix length plus one sample, then channel equalization is trivial and perfect equalization can be achieved. The channel can be inverted in the frequency domain after a discrete Fourier transform (DFT) by a single complex multiply for each sub-channel. This is usually referred to as frequency-domain equalization (FEQ). Equalization using a traditional FEQ can be expressed mathematically in a matrix format as:{circumflex over (X)}i=F·Yi 
where Yi is the vertical DFT output vector of length N at time i (i.e., the tones) and F is the diagonal matrix of FEQ taps (the last N/2−1 elements of Yi contain redundant information and may be ignored).
In practice, the length of the channel impulse response is often much longer than the cyclic prefix length. This results in inter-symbol interference (ISI) and inter-channel interference (ICI) that reduces the signal-to-noise ratio (SNR) that is achieved. This is almost always the case for ADSL systems. The most common way to equalize the channel in these cases is to use a time-domain equalizer (TEQ) to perform channel shortening. Many different methods can be used for computing the TEQ coefficients. Using a TEQ can significantly improve channel equalization, but rarely eliminates ISI completely. The lower band-edge of the downstream channel in ADSL systems often proves particularly difficult to equalize only using a TEQ.
Alternative equalization methods such as a generalized decision-feedback equalizer (GDFE) can be used for channel equalization of multi-carrier systems such as DMT, but the complexity may be very large when compared to TEQ/DFT/FEQ equalization.